SQUARE ROOT 



5515 



SQUARE ROOT 



Many numbers of this class should be squared 

 on cross-section paper by students taking up 

 the subject of square root. 



The root is found by taking the square apart 

 along the lines upon which it was built up. In 

 other words, it must be taken apart along its 

 construction lines. 



Square Root of Numbers Consisting of 10's 

 and Units. What number squared gives 2025? 

 n2= 2025 

 n = v/2025 

 2025 = tenss + 2 X tens X units + units2 



40 + 5 

 2025 



= tens2 



2X40 = 



4 2 5 = 2 X tens X units + units2 

 400 = 2 x tens X units 



25 = units2 



25=units2 



Explanation : 



(1) What is the largest square of tens in 

 2025? 1600. 



(2) What, then, is the tens? 40. 



(3) Place 40, or 4 tens, in the root. 



(4) Subtract 1600; the remainder 425 must 

 contain 2 X tens Xunits+ units 2 . 



(5) Units 2 , being comparatively small, may 

 be neglected for the time; and 425=2 X tens X 

 units. 



(6) The tens is 40; and 425=2 X 40 X units. 



(7) Divide 425 by 2X40 and find units to 

 be 5. 



(8) Subtract 5X80 from 425. 



(9) The remainder must be units 2 . 

 (10) It is found to be so 25=5 2 . 



When dividing by 2 X tens, we must bear in 

 mind that there must be a remainder equal 

 to the units 2 . The following problem illus- 

 trates the point: 



20 + 8 = 28 



2X20 = 40 



320 



64 = 82 



Note that when dividing 384 by 40, it seems 

 that the quotient is 9. But upon taking out 

 9X40 there is not enough left to give units 2 ; 

 the remainder is 384 360 or 24, which is not 

 9 2 . So we see the quotient is only 8, giving 

 a remainder of 64, which is 8 2 . 



Squares of numbers from 10 to 100 have 

 three or four digits; squares of numbers from 

 100 to 1000 have five or six digits. So the 

 square root of any number of three or t four 

 digits lies between 10 and 100; that is, has two 



digits. The square root of a number of five 

 and six digits lies between 100 and 1000; that 

 is, has three digits. Therefore, in finding the 

 square root, an integral number is separated 

 into groups of two digits each, beginning at the 

 right. The number of digits in the root is 

 equal to the number of groups; for example, 

 V72'25 has two digits, Vl'63'84 has three 

 digits, V 10 '49 '76 has three digits. 



A decimal number is divided into groups, 

 beginning at the decimal point, and counting 

 to the left and to the right; for example, 

 V2'07.36, V2'08.22'49. 



The usual concise method of solution is as 

 follows : 



28 



7'84 



4 



2X20 = 40 pTsT 



+ 8 3 84 



48 I 



Here the zeros showing the full value of tens 

 and tens 2 are dropped, and units when found 

 is added to the "trial divisor" (2 X tens) before 

 multiplication, thus including the square of the 

 units in the product. With students taking it 

 up in arithmetic, the first method given here 

 is much more easily understood. 



Square Root of Numbers of More Than Four 

 Digits. Find the square root of 104976. 



\/1049?6 = n 

 300 + 20 + 4 = 324 

 104976 

 90000t2 



2X300 = 6001 14976 



I 12000 = 2XtXU 

 2976 

 400 = u2 



2X320 = 6401 2576 



16 = 42 



After 320 is found, we know there are 32 tens 

 in the root, and after taking out 202, O r 400, we 

 proceed to find units. The process is shortened 

 below : 



2X320 = 640 



4 



644 



Square Root of Decimal Numbers. 1. A 

 product contains as many decimal places as the 

 two factors that make it. 



2. The square is a product and the factors are 

 equal. 



